Network Reduction Theory

This page explains the theory and mathematics behind network reduction. Power systems can have thousands of buses and branches; reduction shrinks the network — making large studies tractable and matrices faster to build and invert — while preserving the power-flow relationships at the retained buses. Which characteristics survive depends on the strategy (RadialReduction, DegreeTwoReduction, WardReduction).

The graph and susceptance structure reduction operates on

Network reduction is fundamentally a graph operation, and to understand why it is well-defined it helps to see the three matrices that encode the network's topology and electrical strength. These are the same building blocks the DC sensitivity matrices are assembled from, and what every reduction manipulates under the hood. (For constructors and accessors see the public API reference; the discussion here is about what they mean.)

The incidence matrix: pure topology

The IncidenceMatrix$A$ is the oriented node–arc incidence matrix: one row per arc, one column per bus, with a $+1$ at the arc's from bus, a $-1$ at its to bus, and zeros elsewhere. It carries topology only — which bus connects to which, and with what orientation — and nothing electrical. The reference-bus column is dropped so the downstream susceptance matrix is non-singular. Reading $A$ column by column recovers each bus's degree, which is exactly the quantity radial (degree 1) and degree-two reductions key off of.

The BA matrix: topology weighted by electrical strength

The BA_Matrix is the product $B A$, where $B$ is the diagonal matrix of branch susceptances ($b = 1/x$ under the DC approximation). Where $A$ says which buses a branch connects, $BA$ scales each connection by how electrically strong it is. Mapped onto bus angles it returns branch flows — the linear operator behind $P_{ij} = (\theta_i - \theta_j)/x_{ij}$.

The ABA matrix: the grounded graph Laplacian

The ABA_Matrix is $A^\top B A$, the reduced nodal susceptance matrix — a weighted graph Laplacian^[laplacian] with the reference bus grounded out. Solving $ABA\,\theta = P$is the DC power flow, and inverting it produces the dense PTDF/LODF sensitivities. Because it is a Laplacian, eliminating a bus is a Kron elimination on this matrix — which is precisely why degree-two reduction (below) is exact rather than approximate.

Why this makes reduction well-posed

Every reduction is defined on this susceptance graph and then propagates uniformly to the downstream matrices (see also Computational considerations, on why reductions are applied to the Ybus first). A radial bus is a degree-1 node; a degree-two bus a degree-2 node; Ward reduction^[ward] Kron-eliminates the external subgraph. Because all are operations on the incidence/susceptance structure, the same reduction map applies to PTDF, LODF, and their virtual variants without re-deriving anything per matrix.

A subtlety: the susceptance graph is not the admittance graph

Connectivity and reduction do not always see the same network. find_subnetworks walks the admittance graph of the Ybus, whereas ABA is built from the susceptance graph. These differ for branches with $r > 0$ and $x = 0$: such a branch has finite admittance but zero susceptance ($b = 1/x$ absent), so it appears in the Ybus graph but not in BA_Matrix. A network that is one connected island electrically can therefore fragment into several components in the susceptance graph, leaving blocks with no reference bus and a singular ABA. This is why zero-impedance handling must resolve both endpoints of such a branch to a common node before building the susceptance matrix.

Radial branch reduction

A radial branch connects to a bus with only one connection to the rest of the network — a "dead-end."

Main Network --- Bus A --- Bus B (radial)
                         \
                          Bus C (radial)

For a radial bus $k$ connected to bus $j$, the DC relation $P_k \approx (\theta_j - \theta_k)/X_{jk}$ gives

\[\theta_k = \theta_j - P_k X_{jk}.\]

The radial-bus angle is completely determined by its parent, so the bus can be eliminated by transferring its load to the parent. Power flows and angles at the core (non-radial) network are preserved exactly; only the eliminated locations lose their explicit representation. Radial reduction is therefore for studies that target the transmission backbone rather than distribution feeders.

Degree-two (Kron) reduction

A degree-two bus connects exactly two others, acting as a pass-through:

Bus A --- Bus B (degree 2) --- Bus C

Kron reduction^[kron] eliminates buses from the admittance matrix. Partitioning into retained ($r$) and eliminated ($e$) buses with no injection at the eliminated buses ($I_e = 0$):

\[Y_{reduced} = Y_{rr} - Y_{re} Y_{ee}^{-1} Y_{er}.\]

For a single degree-two bus $k$ between $i$ and $j$ (no shunt at $k$) this collapses to the series-impedance combination

\[y_{ij}^{new} = y_{ij}^{old} + \frac{y_{ik}\,y_{kj}}{y_{ik} + y_{kj}}.\]

Equivalently, a pass-through bus with $P_k = 0$ has an angle that is the reactance-weighted average of its neighbors, $\theta_k = (X_{kj}\theta_i + X_{ik}\theta_j)/(X_{ik} + X_{kj})$, so replacing it with a direct equivalent branch preserves the flow relationship. Retained-bus flows and overall impedances are preserved; the eliminated bus and its explicit branches are replaced by one equivalent branch. Avoid it where the degree-two bus carries measurements, controls, or a significant shunt.

Ward reduction

Radial and degree-two reduction eliminate individual buses by their local degree. WardReduction instead removes a whole external subsystem at once, keeping only a chosen set of study buses and preserving how the external network responds as seen from them.

Partition the buses into three groups: the study buses to retain, the external buses to eliminate, and the boundary buses — study buses that a retained branch connects directly to the external area. Ward equivalencing is a Kron (Schur-complement) elimination of the external block of the Ybus: with the admittance matrix partitioned into study/boundary ($s$) and external ($e$) parts,

\[Y_{eq} = Y_{ss} - Y_{se}\,Y_{ee}^{-1}\,Y_{es}.\]

The correction term $-Y_{se}Y_{ee}^{-1}Y_{es}$ is fully dense over the boundary buses; it is realized as a set of equivalent branches between boundary buses plus equivalent shunt admittances at them, so the reduced network reproduces the driving-point and transfer behavior the study area would see from the original external network. Unlike degree-two reduction — which is exact for the DC flows it preserves — a Ward equivalent is exact only for the operating state its external injections encode; it is a boundary-matched approximation for other states, which is why the study area is chosen to contain everything of interest.

Ward reduction needs a non-empty boundary: with no branch crossing between study and external areas there is nothing to match against, and the external buses cannot be folded onto the study set by impedance criteria (a degenerate case the implementation flags rather than guessing an equivalent).

Combining reductions

Reductions can be applied in sequence, and order matters — each pass can expose new candidates for the next. Apply RadialReduction before DegreeTwoReduction: removing dead-ends often exposes new degree-two buses.

Eliminating a bus with no injection is clean; a bus carrying load or generation has those devices mapped to a retained bus, and shunt admittances affect the equivalent admittance matrix. RadialReduction and DegreeTwoReduction accept buses to protect from elimination (e.g. injector hosts). Reduction cannot preserve detailed voltage profiles, local dynamics, or exact AC behavior at eliminated locations — keep the full network for state estimation, protection coordination, or local voltage studies.

References